This will require a NG+ run as you'll need 1,920 parts total to upgrade each weapon.
There are weapons only Ellie has access to, and some only for Abby. To upgrade, you take any of the found parts you get and use them at the Workbenches you come across during the game.
Here are each weapon, their owners, and how much to upgrade them all (credit to PowerPyx for the breakdown):
Semi-Auto Pistol - Ellie (170)
- Fire Rate (40)
- Stability (50)
- Recoil (30)
- Capacity (50)
Revolver - Ellie (200)
- Fire Rate (30)
- Stability (50)
- Reload Speed (50)
- Damage (70)
Bow - Ellie (190)
- Draw Speed (50)
- Stability (60)
- Rangefinder (80)
Bolt-Action Rifle - Ellie (280)
- Stability (50)
- Capacity (50)
- 6x Scope (80)
- Damage (100)
Pump Shotgun - Ellie (130)
- Fire Rate (40)
- Stability (40)
- Capacity (50)
Semi-Auto Rifle - Abby (270)
- Stability (50)
- Capacity (60)
- 4x Scope (80)
- Burst Fire (80)
Crossbow - Abby (190)
- Stability (50)
- 4x Scope (60)
- Reload Speed (80)
Double Barrel Shotgun - Abby (130)
- Stability (30)
- Reload Speed (40)
- Damage (60)
Military Pistol - Abby (170)
- Fire Rate (40)
- Stability (50)
- Recoil (30)
- Capacity (50)
Hunting Pistol - Abby (190)
- Stability (50)
- 4x Scope (60)
- Damage (80)
The Flamethrower and the Silenced Sub-machine Gun do not have upgrades.